Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected subgraph G of G with vertex set V whose maximum degree is at most 14 + 2π/γ . If G is the Delaunay triangulation of V , and γ = 2π/3, we show that G is a t-spanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is a triangulation

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... ing the diamond property, then for a specific range of values of γ dependent on the angle of the diamonds, we show that G is a t-spanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on γ. If G is the graph consisting of all Delaunay edges of length at most 1, and γ = π/3, we show that a modified version of the algorithm produces a plane subgraph G of the unit-disk graph which is a t-spanner (for some constant t) of the unit-disk graph of V , whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25. We emphasize that the graph G in Theorem 2 is not necessarily a subgraph of the Delaunay triangulation, but it is a plane subgraph of the unit-disk graph. Bounded degree plane spanners of the unit-disk graph have applications in topology control of wireless ad hoc networks, see [9] . In such a network, a vertex u can only communicate